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Köthe–Toeplitz duals of some vector-valued Orlicz sequence spaces. (English) Zbl 1096.46008
Summary: We obtain the generalized Köthe–Toeplitz duals of some vector-valued Orlicz sequence spaces. We extend the definition of I. J. Maddox [Int. J. Math. Math. Sci. 3, 423–432 (1980; Zbl 0464.46014)] a bit, and find the $$\alpha$$-dual of $$F(X_k,M)$$ defined in [D. Ghosh and P. D. Srivastava, Indian J. Pure Appl. Math. 30, No. 8, 819–826 (1999; Zbl 0935.46008)]. Further, we calculate the $$\beta$$-dual of $$h(X_k,M)$$ and $$\ell(X_k,M)$$, which are the most important special cases of $$F(X_k,M)$$. Several Orlicz sequence spaces such as $$\ell_M$$ and $$\ell_M(X)$$ are only special cases of $$\ell(X_k,M)$$ as well as of $$F(X_k,M)$$.
##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 46A20 Duality theory for topological vector spaces 46B45 Banach sequence spaces
##### Keywords:
Köthe-Toeplitz dual; vector-valued Orlicz space